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This is exercise 70, chapter 4. from Folland (page 142)

Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in C(X, \mathbb{R})$ then $fg\in J$.

  1. If $J$ is an ideal in $C(X, \mathbb{R})$, let $h(J) = \{x \in X: f(x) = 0,\ \forall f \in J\}$. Then $h(J)$ is a closed subset of $X$, called the hull of $J$.
  2. If $E\subset X$, let $k(E)=\{f \in C(X, \mathbb{R}) : f(x)=0,\ \forall x \in E\}$. Then $k(E)$ is a closed ideal in $C(X, \mathbb{R})$, called the kernel of $E$.
  3. If $E\subset X$, then $h(k(E)) =\overline{E}$.
  4. If $J$ is an ideal in $C(X, \mathbb{R})$ then $k(h(J))=\overline{J}$. (Hint: $k(h(J))$ may be identified with a subalgebra of $C_0(U, \mathbb{R})$ where $U=X\setminus h(J)$)

I've managed to prove assignments 1-3. In (4)

if $f$ is from $J$ then for each $y\in h(J)$ is $f(y)=0$, then must be $f\in k(h(J))$. Since $k(h(J))$ is closed, also $\overline J\subset k(h(J))$.

For other way around, I've proven the hint ($k(h(J))$ may be identified with a subalgebra of $C_0(U, \mathbb{R})$ where $U=X\setminus h(J)$), using some corollary fo Stone-Weierstrass), but don't understand how to make conection between $C_0(X\setminus h(J), \mathbb{R})$ and $\overline{J}$

ANY HELP WOULD BE APPRECIATED.

P.S. This is my first time asking question, but I've found this site very helpfull, so thanks!

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1 Answer

I don't know for the moment how to use the hint. However, we can try the following.

  1. Fix $f\in k(h(J))$ and $\varepsilon>0$. Let $K:=\{x\in X, |f(x)|\geqslant \varepsilon\}$. By compactness, we can find $g\in J$ non-negative such that $g>0$ on $K$. Indeed, $K$ is compact, and the intersection $\bigcap_{g\in J}g^{-1}(\{0\})\cap K$ is empty, so there are $g_1,\dots,g_n$ such that $K\cap\bigcap_{j=1}^ng_j^{-1}(\{0\})\cap K$ is empty. Then take $g:=\sum_{j=1}^ng_j^2$.

  2. Let $f_n(x):=\frac{f(x)}{\frac 1n+g(x)}g(x)$. It's an element of $J$ such that $\lVert f_n-f\rVert\leqslant 2\varepsilon$.

It's also exercise 1.5.b) in Elements of Functional Analysis, by F. Hirsch and G. Lacombe, Springer

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First, forgive me for taking so long to come to answer. I can't figure out why function g exists. –  Ana Apr 11 '13 at 10:34
    
@ana I've added details. –  Davide Giraudo Apr 11 '13 at 11:38
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